Optimal Control Approaches for Some Geometric Optimization Problems

This work is a survey on optimal control methods applied to shape optimization problems. The unknown character of the domain where the state system is defined creates major difficulties both for the theoretical analysis and for the implementation of the numerical approximation methods. The optimal control and the controllability methods for elliptic equations allow the development of approximation procedures defined in a given domain that contains all the admissible subdomains for the original optimal design problem. This class of methods enters the so-called fixed domain methods that have numerous advantages in the treatment of unknown/variable domains problems. Such methods are very much studied in the scientific literature, both in the case of geometric optimization problems and in the case of free boundary problems.